Consider a colloidal suspension of latex particles confined between two plates. The plates are charged, so there is an electric potential difference V between them (as shown in the figure below). The number of particles is N and the temperature is T. The system is isolated from the environment and gravitational effects are negligible.
Now each particle carries a positive charge +q, so particles are more likely to be near the plate at the top in the figure, which is negatively charged, than the bottom one, which is positively charged. The system is very dilute, so particles are independent of each other and the electrostatic interaction between particles can be neglected. For simplicity, assume that in the z direction each particle may only be at one of the following two locations: either attached to the top plate, or attached to the bottom one, as shown in the figure.
(i) Calculate the fraction of particles attached to the top plate. Sketch this as a function of: (a) T, (b) V. Discuss the physical meaning of your results.
Hint for (i): The total energy of the system, E, depends on how the particles are distributed between the two plates. Find out the number of configurations associated with a certain E and express the entropy of the system, S, as a function of E, V, and N. Then use T = ∂E/∂S|V,N and solve for E.
(ii) Consider the case at room temperature (300±K) and q = 10 e, where e is the magnitude of the charge carried by an electron (1.6µ10-19 C). (This is due to the ionization of 10 charged groups chemically attached to the particle surface.) How large does V need to be so that the particles have a significant preference to reside at the top plate? (In reality, there is salt in the solution, and the salt ions interact with the particles in a complex way and partially neutralize them. So q here should be regarded as the effective charge carried by a particle after considering these effects.)